Question

Solve the equation x+3/x-4 = x-5/x+4

Answer 1

`\large\boxed{x=(1)/(2)}`

Explanation:

`(x+3)/(x-4)=(x-5)/(x+4)\\\\\text{Domain:}\ x-4\\eq0\ \wedge\ x+4\\eq0\\\\x\\eq4\ \wedge\ x\\eq-4\\\\D:x\in\mathbb{R}-\{-4,\ 4\}`

`(x+3)/(x-4)=(x-5)/(x+4)\qquad\text{cross multiply}\\\\(x+3)(x+4)=(x-4)(x-5)\qquad\text{use FOIL}\ (a+b)(c+d)=ac+ad+bc+bd\\\\(x)(x)+(x)(4)+(3)(x)+(3)(4)=(x)(x)+(x)(-5)+(-4)(x)+(-4)(-5)\\\\x^2+4x+3x+12=x^2-5x-4x+20\qquad\text{combine like terms}\\\\x^2+(4x+3x)+12=x^2+(-5x-4x)+20\\\\x^2+7x+12=x^2-9x+20\qquad\text{subtract}\ x^2\ \text{from both sides}\\\\7x+12=-9x+20\qquad\text{subtract 12 from both sides}\\\\7x=-9x+8\qquad\text{add}\ 9x\ \text{to both sides}\\\\16x=8\qquad\text{divide both sides by 16}\\\\x=(8)/(16)\\\\x=(8:8)/(16:8)\\\\x=(1)/(2)\in D`